Incremental Decomposition Methods

Description

Incremental Multiple Correspondence Analysis and Principal Component Analysis

Details

Author(s)

Alfonso Iodice D' Enza [aut], Angelos Markos [aut, cre], Davide Buttarazzi [ctb]

References

Hall, P., Marshall, D., & Martin, R. (2002). Adding and subtracting eigenspaces with eigenvalue decomposition and singular value decomposition. Image and Vision Computing, 20(13), 1009-1016.

Iodice D' Enza, A., & Markos, A. (2015). Low-dimensional tracking of association structures in categorical data, Statistics and Computing, 25(5), 1009–1022.

Iodice D'Enza, A., Markos, A., & Buttarazzi, D. (2018). The idm Package: Incremental Decomposition Methods in R. Journal of Statistical Software, Code Snippets, 86(4), 1–24. DOI: 10.18637/jss.v086.c04.

Ross, D. A., Lim, J., Lin, R. S., & Yang, M. H. (2008). Incremental learning for robust visual tracking. International Journal of Computer Vision, 77(1-3), 125–141.

Adds two eigenspaces using block-wise incremental SVD (with or without mean update)

Description

This function implements two procedures for updating existing decomposition. When method="esm" it adds two eigenspaces using the incremental method of Hall, Marshall & Martin (2002). The results correspond to the eigenspace of the mean-centered and concatenated data. When method = "isvd" it adds the eigenspace of an incoming data block to an existing eigenspace using the block-wise incremental singular value decomposition (SVD) method described by Zha & Simon (1999), Levy and Lindenbaum (2000), Brand (2002) and Baker (2012). New data blocks are added row-wise. The procedure can optionally keep track of the data mean using the orgn argument, as described in Ross et al. (2008) and Iodice D'Enza & Markos (2015).

Usage

add_es(eg, eg2, current_rank, ff = 0, method = c("esm", "isvd"))

Arguments

(*) for method = "esm" only; (**) for method = "isvd" only.

Value

A list describing the SVD of a data matrix, with components

References

Zha, H., & Simon, H. D. (1999). On updating problems in latent semantic indexing. SIAM Journal on Scientific Computing, 21(2), 782-791.

Levy, A., & Lindenbaum, M. (2000). Sequential Karhunen-Loeve basis extraction and its application to images. IEEE Transactions on Image Processing, 9(8), 1371-1374.

Brand, M. (2002). Incremental singular value decomposition of uncertain data with missing values. In Computer Vision-ECCV 2002 (pp. 707-720). Springer Berlin Heidelberg.

Ross, D. A., Lim, J., Lin, R. S., & Yang, M. H. (2008). Incremental learning for robust visual tracking. International Journal of Computer Vision, 77(1-3), 125-141.

Baker, C. G., Gallivan, K. A., & Van Dooren, P. (2012). Low-rank incremental methods for computing dominant singular subspaces. Linear Algebra and its Applications, 436(8), 2866-2888.

Iodice D' Enza, A., & Markos, A. (2015). Low-dimensional tracking of association structures in categorical data, Statistics and Computing, 25(5), 1009-1022. Iodice D'Enza, A., Markos, A., & Buttarazzi, D. (2018). The idm Package: Incremental Decomposition Methods in R. Journal of Statistical Software, Code Snippets, 86(4), 1–24. DOI: 10.18637/jss.v086.c04.

See Also

do_es, i_pca, i_mca, update.i_pca, update.i_mca

Examples

## Example 1 - eigenspace merge (Hall et al., 2002)
#Iris species
data("iris", package = "datasets")
X = iris[,-5]
#obtain two eigenspaces
eg = do_es(X[1:50, ])
eg2 = do_es(X[c(51:150), ])
#add the two eigenspaces keeping track of the data mean
eg12 = add_es(method = "esm", eg, eg2)
#equivalent to the SVD of the mean-centered data (svd(scale(X, center = TRUE,scale = FALSE)))

## Example 2 - block-wise incremental SVD with mean update, full rank (Ross et al., 2008)
data("iris", package = "datasets")
# obtain the eigenspace of the first 50 Iris species
X = iris[,-5]
eg = do_es(X[1:50, ])
#update the eigenspace of the remaining species to
eg_new = add_es(method = "isvd", eg, data.matrix(X[c(51:150), ]))
#equivalent to the SVD of the mean-centered data (svd(scale(X, center = TRUE, scale = FALSE)))

##Example 3 - incremental SVD with mean update, 2d approximation (Ross et al., 2008)
data("iris", package = "datasets")
# obtain the eigenspace of the first 50 Iris species
X = iris[,-5]
eg = do_es(X[1:50, ])
#update the eigenspace of the remaining species to
eg = add_es(method = "isvd", eg, data.matrix(X[c(51:150), ]),current_rank = 2)
#similar to PCA on the covariance matrix of X (SVD of the mean-centered data)

Computes the eigenspace of a data matrix

Description

This function computes the eigenspace of a mean-centered data matrix

Usage

do_es(data)

Arguments

Value

A list describing the eigenspace of a data matrix, with components

See Also

add_es, update.i_pca, i_pca

Examples

#Iris species
data("iris", package = "datasets")
eg = do_es(iris[,-5]) 
#corresponds to the SVD of the centered data matrix

enron data set

Description

The data set is a subset of the Enron e-mail corpus from the UCI Machine Learning Repository (Lichman, 2013). The original data is a collection of 39,861 email messages with roughly 6 million tokens and a 28,102 term vocabulary. The subset is a binary (presence/absence) data set containing the 80 most frequent words which appear in the original corpus.

Usage

data("enron")

Format

A binary data frame with 39,861 observations (e-mail messages) on 80 variables (words).

References

Lichman, M. (2013). UCI Machine Learning Repository [http://archive.ics.uci.edu/ml]. Irvine, CA: University of California, School of Information and Computer Science.

Examples

data(enron)

Incremental Multiple Correspondence Analysis (MCA)

Description

This function computes the Multiple Correspondence Analysis (MCA) solution on the indicator matrix using two incremental methods described in Iodice D'Enza & Markos (2015)

Usage

i_mca(data1, data2, method=c("exact","live"), current_rank, nchunk = 2, 
 ff = 0, disk = FALSE)

Arguments

Value

References

Hall, P., Marshall, D., & Martin, R. (2002). Adding and subtracting eigenspaces with eigenvalue decomposition and singular value decomposition. Image and Vision Computing, 20(13), 1009-1016.

Iodice D' Enza, A., & Markos, A. (2015). Low-dimensional tracking of association structures in categorical data, Statistics and Computing, 25(5), 1009–1022.

Iodice D'Enza, A., Markos, A., & Buttarazzi, D. (2018). The idm Package: Incremental Decomposition Methods in R. Journal of Statistical Software, Code Snippets, 86(4), 1–24. DOI: 10.18637/jss.v086.c04.

Ross, D. A., Lim, J., Lin, R. S., & Yang, M. H. (2008). Incremental learning for robust visual tracking. International Journal of Computer Vision, 77(1-3), 125–141.

See Also

update.i_mca, i_pca, update.i_pca, add_es

Examples

##Example 1 - Exact case
data("women", package = "idm")
nc = 5 # number of chunks
res_iMCAh = i_mca(data1 = women[1:300,1:7], data2 = women[301:2107,1:7]
,method = "exact", nchunk = nc)
#static MCA plot of attributes on axes 2 and 3
plot(x = res_iMCAh, dim = c(2,3), what = c(FALSE,TRUE), animation = FALSE)

#\donttest is used here because the code calls the saveLatex function of the animation package 
#which requires ImageMagick or GraphicsMagick and 
#Adobe Acrobat Reader to be installed in your system 
#Creates animated plot in PDF for objects and variables
plot(res_iMCAh, animation = TRUE, frames = 10, movie_format = 'pdf')


##Example 2 - Live case
data("tweet", package = "idm")
nc = 5
#provide attributes with custom labels
labels = c("HLTN", "ICN", "MRT","BWN","SWD","HYT","CH", "-", "-/+", "+", "++", "Low", "Med","High")
#mimics the 'live' MCA implementation 
res_iMCAl = i_mca(data1 = tweet[1:100,], data2 = tweet[101:1000,],
method="live", nchunk = nc, current_rank = 2)


#\donttest is used here because the code calls the saveLatex function of the animation package 
#which requires ImageMagick or GraphicsMagick and 
#Adobe Acrobat Reader to be installed in your system 
#See help(im.convert) for details on the configuration of ImageMagick or GraphicsMagick.
#Creates animated plot in PDF for observations and variables
plot(res_iMCAl, labels = labels, animation = TRUE, frames = 10, movie_format = 'pdf')

Incremental Principal Component Analysis (PCA)

Description

This function computes the Principal Component Analysis (PCA) solution on the covariance matrix using the incremental method of Hall, Marshall & Martin (2002).

Usage

i_pca(data1, data2, current_rank, nchunk = 2, disk = FALSE)

Arguments

Value

References

Hall, P., Marshall, D., & Martin, R. (2002). Adding and subtracting eigenspaces with eigenvalue decomposition and singular value decomposition. Image and Vision Computing, 20(13), 1009-1016.

Iodice D' Enza, A., & Markos, A. (2015). Low-dimensional tracking of association structures in categorical data, Statistics and Computing, 25(5), 1009–1022.

Iodice D'Enza, A., Markos, A., & Buttarazzi, D. (2018). The idm Package: Incremental Decomposition Methods in R. Journal of Statistical Software, Code Snippets, 86(4), 1–24. DOI: 10.18637/jss.v086.c04.

See Also

update.i_pca, i_mca, update.i_mca, add_es

Examples

data("segmentationData", package = "caret")
#center and standardize variables, keep 58 continuous attributes
HCS = data.frame(scale(segmentationData[,-c(1:3)]))
#abbreviate variable names for plotting
names(HCS) = abbreviate(names(HCS), minlength = 5)
#split the data into starting data and incoming data
data1 = HCS[1:150, ]
data2 = HCS[151:2019, ]
#Incremental PCA on the HCS data set: the incoming data is 
#splitted into twenty chunks; the first 5 components/dimensions 
#are computed in each update
res_iPCA = i_pca(data1, data2, current_rank = 5, nchunk = 20)
#Static plots 
plot(res_iPCA, animation = FALSE)

#\donttest is used here because the code calls the saveLatex function of the animation package 
#which requires ImageMagick or GraphicsMagick and 
#Adobe Acrobat Reader to be installed in your system 
#See help(im.convert) for details on the configuration of ImageMagick or GraphicsMagick.
#Creates animated plot in PDF for objects and variables
plot(res_iPCA, animation = TRUE, frames = 10, movie_format = 'pdf')


#Daily Closing Prices of Major European Stock Indices, 1991-1998 
data("EuStockMarkets", package = "datasets") 
res_iPCA = i_pca(data1 = EuStockMarkets[1:50,], data2 = EuStockMarkets[51:1860,], nchunk = 5) 

#\donttest is used here because the code calls the saveLatex function of the animation package 
#which requires ImageMagick or GraphicsMagick and 
#Adobe Acrobat Reader to be installed in your system 
#See help(im.convert) for details on the configuration of ImageMagick or GraphicsMagick.
#Creates animated plot in PDF movies for objects and variables
plot(res_iPCA, animation = TRUE, frames = 10, movie_format = 'pdf')

Plotting 2D maps in Multiple Correspondence Analysis

Description

Graphical display of Multiple Correspondence Analysis results in two dimensions

Usage

## S3 method for class 'i_mca'
plot(x, dims = c(1,2), what = c(TRUE,TRUE), 
contrib = "none", dataname = NULL, labels = NULL, animation = TRUE, 
frames = 10, zoom = TRUE, movie_format = "gif", binary = FALSE,...)

Arguments

Details

The function plot.i_mca makes a two-dimensional map of the object created by i_mca with respect to two selected dimensions. In this map both the row and column points are scaled to have inertias (weighted variances) equal to the principal inertia (eigenvalue or squared singular value) along the principal axes, that is both rows and columns are in pricipal coordinates.

References

Greenacre, M.J. (1993) Correspondence Analysis in Practice. London: Academic Press.
Greenacre, M.J. (1993) Biplots in Correspondence Analysis, Journal of Applied Statistics, 20, 251-269.
ImageMagick: http://www.imagemagick.org; GraphicsMagick: http://www.graphicsmagick.org

See Also

plot.i_pca

Examples

data("women", package = "idm")
res_iMCAl = i_mca(data1 = women[1:50, 1:4], data2 = women[51:300, 1:4], 
method = "live", nchunk = 4)
#static plot, final solution
plot(res_iMCAl, contrib = "ctr", animation = FALSE)

#\donttest is used here because the code calls the saveLatex function of the animation package 
#which requires ImageMagick or GraphicsMagick and 
#Adobe Acrobat Reader to be installed in your system 
#See help(im.convert) for details on the configuration of ImageMagick or GraphicsMagick.
#Creates animated plots in PDF for objects and variables
plot(res_iMCAl, contrib = "ctr", animation = TRUE, frames = 10, movie_format = 'pdf')

Plotting 2D maps in Principal Component Analysis

Description

Graphical display of Principal Component Analysis results in two dimensions

Usage

## S3 method for class 'i_pca'
plot(x, dims = c(1,2), what = c(TRUE,TRUE), 
dataname = NULL, labels = NULL, animation = TRUE, frames = 10, 
zoom = TRUE, movie_format = "gif", ...)

Arguments

Details

The function plot.i_pca makes a two-dimensional map of the object created by i_pca with respect to two selected dimensions.

References

ImageMagick: http://www.imagemagick.org; GraphicsMagick: http://www.graphicsmagick.org

See Also

plot.i_mca

Examples

data("iris", package = "datasets")
#standardize variables
X = scale(iris[,-5])
res_iPCA = i_pca(data1 = X[1:50,-5], data2 = X[51:150,-5], nchunk = c(50,50))
#static plot, final solution
plot(res_iPCA, animation = FALSE)

##\donttest is used here because the code calls the saveLatex function of the animation package 
#which requires ImageMagick or GraphicsMagick and 
#Adobe Acrobat Reader to be installed in your system 
#See help(im.convert) for details on the configuration of ImageMagick or GraphicsMagick.
#Creates animated plots in PDF for objects and variables
plot(res_iPCA, animation = TRUE, frames = 10, movie_format = 'pdf')

twitter data set

Description

The data set refers to a small corpus of messages or tweets mentioning seven major hotel brands. It was gathered by continuously querying and archiving the Twitter Streaming API service, using the twitteR package in R. A total of 7,296 tweets were extracted within a time period of 6 days, from June 23th to June 28th 2013. Only tweets in the English language were considered. A sentiment polarity variable was calculated, indicating the sentiment value of each message and a third variable, user visibility or popularity, as measured by the number of followers each user had, was also included in the dataset

Usage

data("tweet")

Format

A data frame with the following variables:

Brand

The hotel brand mentioned in the tweet: 1=Hilton, 2=Intercontinental, 3=Marriott, 4=Bestwestern, 5=Starwood, 6=Hyatt, 7=Choice

Sentiment

Sentiment for each tweet: 1=negative (-), 2=mixed (+/-), 3=positive (+), 4=very positive (++)

UserVis

User popularity/visibility in Twitter: 1=low, 2=medium, 3=high

References

Iodice D' Enza, A., & Markos, A. (2015). Low-dimensional tracking of association structures in categorical data, Statistics and Computing, 25(5), 1009-1022.

Iodice D'Enza, A., Markos, A., & Buttarazzi, D. (2018). The idm Package: Incremental Decomposition Methods in R. Journal of Statistical Software, Code Snippets, 86(4), 1–24. DOI: 10.18637/jss.v086.c04.

Examples

data(tweet)

Updates a Multiple Correspondence Analysis solution

Description

This function updates the Multiple Correspondence Analysis (MCA) solution on the indicator matrix using the incremental method of Ross, Lim, Lin, & Yang (2008)

Usage

## S3 method for class 'i_mca'
update(object, incdata, current_rank, ff = 0, ...)

Arguments

Value

References

Iodice D'Enza, A., Markos, A., & Buttarazzi, D. (2018). The idm Package: Incremental Decomposition Methods in R. Journal of Statistical Software, Code Snippets, 86(4), 1–24. DOI: 10.18637/jss.v086.c04.

Ross, D. A., Lim, J., Lin, R. S., & Yang, M. H. (2008). Incremental learning for robust visual tracking. International Journal of Computer Vision, 77(1-3), 125–141.

See Also

add_es, i_mca, plot.i_mca

Examples

data(women, package = "idm")
dat = women[,c(1:4)]
res_MCA = i_mca(dat[1:300,])
aa = seq(from = 301, to = nrow(women), by = 200)
aa[length(aa)] = nrow(dat)+1
for (k in c(1:(length(aa)-1)))
{
  res_MCA = update(res_MCA,dat[c((aa[k]):(aa[k+1]-1)),])
}
plot(res_MCA, what = c(FALSE, TRUE), animation = FALSE)

Updates a Principal Component Analysis solution

Description

This function updates the Principal Component Analysis (PCA) solution on the covariance matrix using the incremental method of Hall, Marshall & Martin (2002)

Usage

## S3 method for class 'i_pca'
update(object, incdata, current_rank, ...)

Arguments

Value

References

Hall, P., Marshall, D., & Martin, R. (2002). Adding and subtracting eigenspaces with eigenvalue decomposition and singular value decomposition. Image and Vision Computing, 20(13), 1009-1016.

Iodice D' Enza, A., & Markos, A. (2015). Low-dimensional tracking of association structures in categorical data, Statistics and Computing, 25(5), 1009–1022.

Iodice D'Enza, A., Markos, A., & Buttarazzi, D. (2018). The idm Package: Incremental Decomposition Methods in R. Journal of Statistical Software, Code Snippets, 86(4), 1–24. DOI: 10.18637/jss.v086.c04.

See Also

update.i_mca, i_pca, i_mca, add_es

Examples

data(segmentationData, package = "caret")
HCS = data.frame(scale(segmentationData[,-c(1:3)]))
names(HCS) = abbreviate(names(HCS), minlength = 5)
res_PCA = i_pca(HCS[1:200, ])
aa = seq(from = 201, to = nrow(HCS), by = 200)
aa[length(aa)] = nrow(HCS)+1
for (k in c(1:(length(aa)-1))){
     res_PCA = update(res_PCA, HCS[c((aa[k]):(aa[k+1]-1)),])
    }
#Static plot
plot(res_PCA, animation = FALSE)

women data set

Description

The data are from the third Family and Changing Gender Roles survey conducted in 2002. The questions retained are those related to working women in Spain and the effect on the family. A total of 2,107 respondents answered eight questions on a 5-point Likert scale, as well as four demographic variables (gender, martial status, education and age). There are no cases with missing data.

Usage

data("women")

Format

A data frame with the following variables:

A

"a working mother can establish a warm relationship with her child"
1=strongly agree, 2=agree, 3=neither agree or disagree, 4=disagree, 5=strongly disagree

B

"a pre-school child suffers if his or her mother works"
1=strongly agree, 2=agree, 3=neither agree or disagree, 4=disagree, 5=strongly disagree

C

"when a woman works the family life suffers"
1=strongly agree, 2=agree, 3=neither agree or disagree, 4=disagree, 5=strongly disagree

D

"what women really want is a home and kids"
1=strongly agree, 2=agree, 3=neither agree or disagree, 4=disagree, 5=strongly agree

E

"running a household is just as satisfying as a paid job"
1=strongly agree, 2=agree, 3=neither agree or disagree, 4=disagree, 5=strongly disagree

F

"work is best for a woman's independence"
1=strongly agree, 2=agree, 3=neither agree or disagree, 4=disagree, 5=strongly disagree

G

"a man's job is to work; a woman's job is the household"
1=strongly agree, 2=agree, 3=neither agree or disagree, 4=disagree, 5=strongly disagree

H

"working women should get paid maternity leave"
1=strongly agree, 2=agree, 3=neither agree or disagree, 4=disagree, 5=strongly disagree

g

gender: 1=male, 2=female

m

marital status: 1=married/living as married, 2=widowed, 3=divorced, 4=separated, but married, 5=single, never married

e

education: 1=no formal education, 2=lowest education, 3=above lowest education, 4=highest secondary completed, 5=above higher secondary level, below full university, 6=university degree completed

a

age: 1=16-25 years, 2=26-35, 3=36-45, 4=46-55, 5=56-65, 6=66 and older

Source

http://www.econ.upf.edu/~michael/women_Spain2002_original.xls

References

Greenacre, M. J. (2010). Biplots in practice. Fundacion BBVA.

Examples

data(women)